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Magazine Chapter 12: Problem 18
Find the vertices, foci, and asymptotes of the hyperbola, and sketch its graph. $$ x^{2}-2 y^{2}=3 $$
Short Answer
Expert verified
Vertices: \((\pm \sqrt{3}, 0)\); Foci: \((\pm \sqrt{4.5}, 0)\); Asymptotes: \(y = \pm \frac{1}{2}x\).
Step by step solution
01
Write the Equation in Standard Form
The given equation is \(x^{2} - 2y^{2} = 3\). A hyperbola has a standard form of \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). To convert the given equation into this form, divide everything by 3: \(\frac{x^2}{3} - \frac{2y^2}{3} = 1\). This rearranges the equation to \(\frac{x^2}{\frac{3}{1}} - \frac{y^2}{\frac{3}{2}} = 1\).
02
Identify the Values of a and b
Compare the equation \(\frac{x^2}{3} - \frac{y^2}{1.5} = 1\) with the standard form to identify \(a^2 = 3\) and \(b^2 = 1.5\). Hence, \(a = \sqrt{3}\) and \(b = \sqrt{1.5}\).
03
Find the Vertices
For the hyperbola \(\frac{x^2}{3} - \frac{y^2}{1.5} = 1\), the vertices are located at \((\pm a, 0)\). Substitute \(a = \sqrt{3}\) into the vertices formula: the vertices are \((\pm \sqrt{3}, 0)\).
04
Find the Foci
The foci of a hyperbola in the form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) are at \((\pm c, 0)\), where \(c^2 = a^2 + b^2\). Thus, \(c^2 = 3 + 1.5 = 4.5\), so \(c = \sqrt{4.5}\). Therefore, the foci are at \((\pm \sqrt{4.5}, 0)\).
05
Calculate the Asymptotes
The equations for the asymptotes of a hyperbola \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) are \(y = \pm \frac{b}{a}x\). Here, \(b = \sqrt{1.5}\) and \(a = \sqrt{3}\), so \(\frac{b}{a} = \frac{\sqrt{1.5}}{\sqrt{3}} = \frac{1.5}{3} = \frac{1}{2}\), yielding the asymptotes \(y = \pm \frac{1}{2}x\).
06
Sketch the Hyperbola
Draw the coordinate axes. Mark the vertices at \((\pm \sqrt{3}, 0)\) and the foci at \((\pm \sqrt{4.5}, 0)\). Draw the asymptotes passing through the origin with slopes \(\pm \frac{1}{2}\). Sketch the branches of the hyperbola opening left and right, approaching these asymptotes.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding the Vertices of a Hyperbola
The vertices of a hyperbola are fundamental points where the graph intersects its transverse axis. In the horizontal hyperbola, the vertices give us a sense of the hyperbola's width and orientation. For a hyperbola with an equation in the form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), the vertices are located at \((\pm a, 0)\).
In the given problem, the equation of the hyperbola \(\frac{x^2}{3} - \frac{y^2}{1.5} = 1\) identifies \(a^2 = 3\), leading us to \(a = \sqrt{3}\). Therefore, we determine the vertices to be \((\pm \sqrt{3}, 0)\). These points depict how far the hyperbola stretches along the x-axis from the center. By marking these vertices on a coordinate plane, one can easily visualize the starting points of the hyperbola's branches.
In the given problem, the equation of the hyperbola \(\frac{x^2}{3} - \frac{y^2}{1.5} = 1\) identifies \(a^2 = 3\), leading us to \(a = \sqrt{3}\). Therefore, we determine the vertices to be \((\pm \sqrt{3}, 0)\). These points depict how far the hyperbola stretches along the x-axis from the center. By marking these vertices on a coordinate plane, one can easily visualize the starting points of the hyperbola's branches.
Locating the Foci of a Hyperbola
The foci of a hyperbola, denoted at \((\pm c, 0)\) for a horizontal hyperbola, are fixed points that have crucial geometric significance, always lying further from the center than the vertices. They provide an understanding of the hyperbola's "opening" strength.
For the equation \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), the distance to the foci from the center \(c\) is calculated using the formula \(c^2 = a^2 + b^2\). Given our problem statement, \(a^2 = 3\) and \(b^2 = 1.5\), resulting in \(c^2 = 4.5\). From here, \(c = \sqrt{4.5}\), thus placing the foci at \((\pm \sqrt{4.5}, 0)\).
Positioning the foci on a graph helps to determine the directionality and extent of the hyperbola's branches, defining the shape beyond merely the vertices.
For the equation \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), the distance to the foci from the center \(c\) is calculated using the formula \(c^2 = a^2 + b^2\). Given our problem statement, \(a^2 = 3\) and \(b^2 = 1.5\), resulting in \(c^2 = 4.5\). From here, \(c = \sqrt{4.5}\), thus placing the foci at \((\pm \sqrt{4.5}, 0)\).
Positioning the foci on a graph helps to determine the directionality and extent of the hyperbola's branches, defining the shape beyond merely the vertices.
Determining the Asymptotes of a Hyperbola
Asymptotes are the imaginary lines that a hyperbola approaches but never quite reaches. For hyperbolas, they act as guides indicating where the branches of the hyperbola will tend without ever crossing them. Knowing the asymptotes is vital because they demonstrate the hyperbola's overall behavior at great distances.
For the equation \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), the asymptotes can be determined with the formula \(y = \pm \frac{b}{a}x\). In the case of the given problem, we have \(b = \sqrt{1.5}\) and \(a = \sqrt{3}\), leading to \(\frac{b}{a} = \frac{\sqrt{1.5}}{\sqrt{3}}\), which simplifies to \(\frac{1}{2}\). Thus, the equations for the asymptotes are \(y = \pm \frac{1}{2}x\).
With these asymptotes, sketched as diagonal lines passing through the origin with slopes of \(\pm \frac{1}{2}\), the trajectory of the hyperbola's branches becomes more predictable and precise, guiding them towards these oblique paths.
For the equation \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), the asymptotes can be determined with the formula \(y = \pm \frac{b}{a}x\). In the case of the given problem, we have \(b = \sqrt{1.5}\) and \(a = \sqrt{3}\), leading to \(\frac{b}{a} = \frac{\sqrt{1.5}}{\sqrt{3}}\), which simplifies to \(\frac{1}{2}\). Thus, the equations for the asymptotes are \(y = \pm \frac{1}{2}x\).
With these asymptotes, sketched as diagonal lines passing through the origin with slopes of \(\pm \frac{1}{2}\), the trajectory of the hyperbola's branches becomes more predictable and precise, guiding them towards these oblique paths.
